Beth two

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(Redirected from 2 to the power of C)

In set theory and other branches of mathematics, (pronounced beth two), or 2^c (pronounced two to the power of c), is a certain cardinal number. It is the 2nd beth number, and is the result of cardinal exponentiation when 2 is raised to the power of c, the cardinality of the continuum.

This number 2^c is the cardinality of many sets, including:

• The power set of the set of real numbers, so it is the number of subsets of the real line, or the number of sets of real numbers;
• The power set of the power set of the set of natural numbers, so it is the number of sets of sets of natural numbers;
• The set of all functions from the real line to itself;
• The power set of the set of all functions from the set of natural numbers to itself, so it is the number of sets of sequences of natural numbers;
• The set of all real-valued functions of n real variables to the real numbers.

Some early set theorists hypothesised the equation

stating that 2^c is equal to the 2nd aleph number. It turns out that the truth of this equation (*) cannot be determined from the standard Zermelo-Fraenkel axioms of set theory; it is true in some models and false in others. (*) is a part of the generalized continuum hypothesis (GCH), but it is possible that (*) is true while the full GCH is false. On the other hand, if (*) is true, then the ordinary continuum hypothesis (CH) must follow, but again it is possible that CH is true while (*) is false.